Multiplicity of Normalized Solutions to a Fractional Logarithmic Schrödinger Equation

Abstract

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation (−Δ)su+V(ϵx)u=λu+ulogu2inRN, under the mass constraint ∫RN|u|2dx=a. Here, N≥2, a,ϵ>0, λ∈R is an unknown parameter, (−Δ)s is the fractional Laplacian and s∈(0,1). We introduce a function space where the energy functional associated with the problem is of class C1. Then, under some assumptions on the potential V and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential V reaches its minimum.

Keywords

• normalized solution
• logarithmic Schrödinger equation
• fractional differential equation
• variational methods